Differential equations of elliptic type with variable operators and general Robin boundary condition in UMD spaces

1.    Introduction and Hypotheses

In a complex Banach space E, we consider the second-order differential equation with variable operator coefficients u" (x) + A (x) u (x) — ши (x) = f (x),    x G (0,1), under the Dirichlet boundary conditions и (1) = U1, and the abstract Robin boundary conditions u' (0) — Hu (0) = do.(3)

Here ш is a positive real number, d ₀ , u 1 are given elements of E , (A (x)) x _ej₀ 1] is a family of closed linear operators whose domains D (A (x)) are dense in E , H is a closed linear operator in E, and f belongs to L p (0,1; E ) where 1 < p <  + ro . This article extends and improves the studies done in [1], where the authors have studied (1) – (3) under the general Robin homogeneous boundary value conditions, in the framework of UMD spaces, where we study the existence, the uniqueness, and the maximal regularity of the classical solution of problem (1) – (3). In particular, we give necessary and sufficient conditions to obtain a unique classical solution of problem (1) – (3) satisfying maximal regularity. We consider some fixed ш ₀ and we set, for ш >  ш ₀ , x G [0,1],

A w (x) = A (x) - ші.

Our aim is to find a classical solution u to (1) – (3), i.e, a function u such that for a.e x G (0,1), u (x) G D (A (x)) and x H A (x) u (x) G Lp (0,1; E)

u G W2p (0,1; E), u satisfies u (0) G D (H) and (1) — (3). Generally, more conditions are needed on f or on E. Here we will assume that

E is a UMD space .

We recall that a Banach space E is UMD if and only if for some p >  1 (and thus for all p ) the Hilbert transform is continuous from L^p ( R ; E ) into itself, see Bourgain [2], Burkholder [3]. Throughout this work we suppose that the family of closed linear operators (A (x)) x _e[₀₁] satisfies

3 w o > 0 , E C > 0 : V x G [0,1] , V z >  0 , (A u ₀ (x) - zI ) ^-1 G L (E) and

^{|| (A} u o ^(x)

zI ) ^{I l}(^e) - 1 + z

This assumption means exactly the ellipticity of our problem in the sense of Krein [4]. It follows that for x G [0,1] , w >  w ₀ the square roots

Qu (x) = - (-A. (x))1/2, are well defined and generate analytic semigroups (eyQ“(x))y>0 not necessarily strongly continuous in 0 see Balakrishnan [5] for dense domains and Martinez-Sanz [6] for non dense domains. We will assume, moreover:

E C >  1 , Ө о G ]0,n[ : V s G R, V x G [0,1] , V w >  w o , ( — A ^ (x)) is G L (E) and

(-Au (x))“ L(E) - СеӨо|s|,

E C , a , ц >  0 : V x, т G [0,1] , V z >  0 , V w >  w ₀ ,

| | A u (x) (A u (x) - zI ) -1 A u (x) -1 - A u (т) -1 ) I 1 with a + 2ц - 2 > 0.

.         C | x - т | a 1 L( E ) - | z + w | *             (8)

This hypothesis is known as the Labbas-Terreni assumption. Operators H and Q _u (x) have to satisfy

E C > 0 : V x G [0,1] , V w >  w o , V ( G E ,

with a + 2ц >  2 and the following commutativity conditions

V x G [0,1] , V w >  w ₀ ,

(Qu (x))-1 (Qu (x) - H)-1 = (Qu (x) - H)-1 (Qu (x))-1, and d0 G D (Q (0)).

Remark 1.

From (6) we deduce that, there exists Ө ₀ E

all x belonging to [0,1] , the resolvent of (А _{Ш 0} (x)) verifies:

] 0, И

and r ₀ >  0 such that for

P (A^ (x)) D Пөо,го = {z E C \ {0} : |argz| < Ө0} U B (O,ro), where B (0,ro) is the closed ball of radius r0 and centered in 0. We denote by Г the boundary of Пө0,r0 oriented from жегӨ0 to ^-e іӨ0.

Equation (1) has been studied by several authors via various approaches. In the constant case of operators A (x) = A, many authors dealt with partial differential equations with non-local boundary conditions. We can first refer to the pioneering works by T. Carleman [7] who in the thirties used singular integral technique to handle an elliptic equation in which boundary values of unknown function on two different points are related. This was the starting point of many studies, for example, [8–11]. The next step was the important paper of Bitsadze and Samarskii [12], in 1969, where the authors analyzed an elliptic equation with unknown functions on the boundary connecting its values at some points on the boundary with other points in the interior of the domain. This problem models some phenomena occurring in plasma physics. Paper [13] gave rise to many works on non-local boundary value problems using different techniques. Let us mention a systematic study done by Skubachevskii [12] and Gurevich [14] and references therein. Yakubov [15] and some others [16, 17] use the operator-differential equation tools to study some classes of elliptic partial differential equations with nonlocal boundary conditions. The Robin condition was treated by M. Cheggag et al [18] in a commutative framework, when f E L p (0,1; E ) with 1 < p <  + ro . They considered that the spectral parameter which appears in the boundary conditions is zero, and gave interesting results for this problem when E is an UMD space where they proved that the problem has a unique classical solution u E W ^{2 ,p} (0,1; E ) П L ^p (0,1; D (A)) such as u (0) E H if and only if d ₀ , u 1 are in the interpolation space (D (A) ,E ) _i₊i _p, 2p 2

(D (A) ,E ) _p respectively. The same authors, in [19], studied the problem (1) - (3), but this time, in the absence of the spectral parameter ^ = 0, in the same commutative frame, in the same commutative setting and in a H¨older space. In other words, they assumed that f belongs to C ^Ө ([0,1] ; E ) with Ө E ]0,1[, and under certain assumptions about the operator A, they studied existence, uniqueness and maximal regularity and then gave some positive results for this problem. They show that the problem (1) – (3) has a unique strict solution u E C ² ([0,1]; E ) П C ([0,1] ; D (A)) such as u (0) E H , satisfying the maximal regularity property u " , Au E C ^Ө ([0,1]; E ) , if and only if u 1 E D (A) , d ₀ E D (Q) and Qd ₀ , f (0), - Au 1 + f (1) are in the interpolation space (D (Q), X) _{1- Ө} ^ , with Q = -/- A.

In the variable case of operators A(x), the commutator hypothesis (8) was used for the first time in Labbas [20] for the same problem but with boundary conditions of Dirichlet type, in Bouziani et al [21] for transmission conditions and Haoua et al [22] and [1] for Robin conditions. All these studies were performed in the frame work of h¨oelderian spaces. For the bounded interval, a direct method based on Dunford’s operational calculus has been used in Labbas [20] under some hypotheses on differentiability of resolvent of operators A^(x). Moreover, the case of differentiability of the resolvent of (A (x))xej0 1] was used by Da Prato and Grisvard [23], Labbas [20] and Boutaous et al [24] . Also, in these studies, the boundary conditions considered were of Dirichlet type. However, in Boutaous et al [24] the authors used the Krein’s approach, under some natural differentiability assumptions on the resolvent of the square roots Qu (x) combining those of Yagi [25] and Acquistapace–Terreni [26].

The organization of the paper is as follows. Section 2 , contains some technical lemmas which will be useful for the study of problem (1) - (3). In Section 3 , an heuristic reasoning is used to obtain a representation of the solution. We obtain an integral equation which is solved using (8). Section 4 is devoted to the study of the maximal regularity of the solution; we give necessary and sufficient compatibility conditions to obtain it. In section 5 , the existence of the solution is proved using the associated approximating problem. Finally, in section 6 , we provide an example to which our abstract results apply.

2.    Technical Lemmas

Lemma 1. There exists C >  0 such that for each z Е Г and r > 0, we have

|z + r| > C |z|, |z + r| > Cr, |z — r| > C |z|, |z — r| > Cr, and

V r > 0 , V v Е [0,1] , [     *dz\    <  C.

J _r | z ± r | | z |       r^v

Proof. See [27, Lemma 6.1 and 6.2, p. 564].

□

Lemma 2. Assume that (6) hold. There exists a constant M >  0 and w * > w ₀ such that, for all w >  w * and x Е [0,1] , operators I ± e ^2Q “ ^(x) are invertible in L (E ) and

(I ± e ^2Q “ ^{(x) ) 1}

≤ M.

L ( E )

Proof. Let x Е [0,1] . Since Qu (x) generates a bounded analytic semigroup and 0 Е P (Qu (x)), there exist M > 1 and 6 > 0 such that for any y > 0 and w > 0, we have l|eyQ“    , . < Me-“.

see [28, Theorem 6.13, p. 74] and in the case of non dense domains see [29, Proposition 2.1.1, p. 35 and Proposition 2.3.1, p. 55,56]. We can choose k ∈ N ^∗ such that

K₁e ^{- 2 kn} 1 ^s <  - < 1.

¹        “ 2

Then I — e ^{2 kQ} ^ ^(x) is boundedly invertible with

( t_ ² k^QQ ( ( x ) \ ¹        /

¹1 e             L ( E ) <  1 — 1/2

so 0 Е p (I — e^{2Q (x )}) since

I

- e2Q^(x)) (i + e2Q“(x) + ... +

_e ²⁽ k ^-i)Q „ ^(x) ) ⁽i

-

e 2 kQ „ ( x ) ) ^-1

-

e 2 kQ „ ( x ) ^{) -1 (}i + e 2 Q „ ( x ) + ... + e 2( k -1) Q „ ( x ) ^{) (}i

-

e 2 Q u ( x ) ⁾ .

Moreover,

< (I + I ' 2 Q IIl

(I e ^2kQ ^ ( x ) ) ¹

+ - + ■    ^{(x) |}Q)

L ( E )

(I

≤

_ _e2kQ „ ( x ) \ ^-1

< 2K k .

L ( E )

We obtain the result for I + e ^2Q “ ^(x) = I _ ( _ e ^2Q “ ^(x) ) if we replace e ^2kQ “ ^(x) , e ^2Q “ ^(x) by _ e ^2kQ “ ^(x) , __eQQx ^(x) in the above proof.

□ Lemma 3. Assume that (6) hold. Then there exist constants C >  0 , w j > w ₀ such that for all w >  w * and x E [0,1] , we have

^{|| (Q w} (^x)     ^zI )    || l(e ) <

C уггм + i z i

Proof. Using [4, p. 116, 117] and for all z >  0 and x E [0,1] , we have

^(Q w ^(x) — ^zI ) = ^_ ^y ^_ A w ^(x) + ^zI ^ =

^_ 1 [ ^{( _} Aw ^{(x) _} AI)     _ ^_ 1 / ' ^ ^s ^{( _} A ^{(x) + wI + sI} ) J

• 2 Jr   zzvx     = “ 7o

Due to hypothesis (6) and Lemma 1, we obtain

|| (Q w (x) _ zI ) ^-1 1

< ^C / o

+^

^{L ( E )} <  ^C Jo ⁺“            t ²

s

(1 + | w | + s) (s + z ² )

(1 + | w | + t ² )(t ² + Z ² )

dt ≤

< C     Г I +“ 1 + | w | _d _ ⁺“        _d 1

- 1 + | w | _ z ² L/_o 1 + | w I + t ^{2 t} J_o t² + z ² J

ds ≤

C

≤ . л/^{1 + | w 1 + | z |}

□

Lemma 4. Assume that (5) - (10) hold. Then there exist constants C >  0 , w * > w ₀ such that for all w >  w * , and x E [0,1], operator Q _w (x) ± H is boundedly invertible and

| (Q . (x) ± H ) - ¹ | _t( E ) <  -C=-

Proof. See [19, Proposition 7, p. 987].

□

For w >  w * , we also define the linear operator Л _w (x) by

Г D (Л w (x)) = D (Q w (x))

I Лw (x) = Qw (x) _ H + e2Q“(x) (Qw (x) + H), x E [0,1], which will be the determinant of the system of our problem.

Lemma 5. Assume (5) - (7) and (10). Then for all w >  w * and x E [0,1] , Л _w (x) is closed and boundedly invertible with

[Лw (x)]-1 = (Qw (x) _ H)- 1 [I + Mw (x)]-1 (I _ e2Q“(x))-1, where

• M . (x) = 2 (I - ■ ' ^(x) ) ^-1 Q . (x) e Q ^(x) (Q . (x) - H ) ^-1 ,

and

[A. (x)]-1 = (Q. (x) - H)-1 + (Q. (x) - H)-1 W (x), with

W (x) eL (E),  (Q. (x) - H)-1 W (x) = W (x) (Q. (x) - H)-1, and

W (x) (E ) c ^Q D ( q . (x) ^k ) .

k =1

Proof. See [22, Lemma 2.5, p. 4].

□

Lemma 6. From (6) and (8), we have

E C, а, ц >  0 : V x, т e [0,1], V z >  0, V w >  w j 11 Q . (x) (Q . (x) - zI ) ^-1 (Q . (x) ^-1 - Q . (т ) ^-1 ) | L _{( E} ) <  with а + 2ц - 2 > 0.

C | x - τ | ^α | z + w | ^

Proof. See [1, Lemma 4 , p. 22 ].

□

3.    Representation of the Solution

Assume that there exists a solution u of (1) - (3 ) satisfying (4). Setting when A . (x) = A - ωI is a constant operator satisfying the natural ellipticity hypothesis is mentioned above (we will take Q . = - ( - A . ) ^1/2 ). By using the method based on the variation of constant and Green’s functions, the solution of problem (1) – (3) is (see [18])

u (x) = e ^xQ “ [Л-Ч + (Q . + H )A,

ω

■'Q u i] + 2e ^xQ“ (Q . + h ) A,

ω

-

2 e

^xQ “ (Q . + H ) A ^-1 e Q “

_{Q .} -1

Q -¹ f Q f (s) ds

J 0

-

^{- A} . ^{1 e} Q d o ]

-

2 e

-

2 e

■ ⁽' s ^Q ' f (s) ds ₊ е ^{(1- х}Ж [(J- - ( q . ₊ h ) A₍

ω

■        u i

-

:Q' (Q . + H ) Л

ω

i _e Q “

_:Q- [i - ( q . ₊ h ) л

ω

1 e 2 Q ^

x

■ ^Q4

] Q -¹ /

Q - ¹ f e ‘^Q " f (s)

-

■ '■ 'Q' f (s) ds ₊

e ^(x-s)Q “ f (s) ds + |Q -¹ X

e ^(s-x)Q . f (s) ds.

Set

L . ( x ) (x, f ) = 2e ^xQ “ ^(x) (Q u (x) + H ) (Л u (x)) ^-1 Q u (x) ^-1 1' e Q ^(x) f (s) ds

-

-

¹ - Q ^{( x )} (Q u (x) + H ) (Л u (x)) ^-1 - Q (AQ - ¹ (x) Г - Q ^{( x )} f (s)

2                                                  Jo

-

-

-

2 -

1 e

_: ^{(1-x)Q u ( x )} (Q u (x) + H ) (Л u (x)) ^-1 ₍ ^{Q ( x )} Q u (x) ^-1 ^ e ^{sQ ( x )} f (s) ds

_:    Q ( x ) ^I - (Q ^ (x) + H ) (Л u (x))

-

¹ e^{2 Q (x)} ] Q -¹ (x) / e^{(1 -}^ ^{Q u ( x )} f (s) + o

+2 Q u (x) ^-1 ^ x -         ^{( x )} f (s) ds + 2 Q u (x) ^-1 ^1 .         ^{( x )} f (s) ds.

We can write that:

^L Q u ( x ) ^{(x, f} ) = ^L Q ( x ) ^(x, u " ^(x) + A u ^(x) u ^(x)) ,

After two integrations by parts and some formal calculus, as in R. Haoua and A. Medeghri [22], we obtain the following abstract equation:

w + PuW = G (x, f), where w (•) = Au (•) u (•) .

Here, for all x G [0,1] , ш >  ш *

(P u w) (x) = ¹ K u (x) .     ^{( x )} f Q u (x) ³ .     ^{( x )} (Q u (s)

2 _o

-2

-

Q u (x) ²) W (s) ds

-

-

-

1 e

2 e

;^{xQ u ( x )} K u (x) e ^{Q ( x )} 0^ Q u (x) ³ e ^(1-s)Q “ ^{(x) (} Q u (s) ,⁽i- x )^Q . ^{( x )} K u (x) -Qu^{( x )} J Q u (x) ³ e ^{sQ ( x )} Q (s)

-2

-2

+2 e

-

+2

-

Q ω

- Q _ω

^{(1 -x ) Q u ( x )} K u (x) е ² ^ ^{( x )} ^ Q u (x) ³ e^- AQ ^{( x )} (Q u (s)

¹ _e ^(1-x)Q ^ ^(x)

j Q u (x) ³ e ^(1-s)Q “ ^(x)

(x) ²) w (s) ds —

(x) ²) w (s) ds +

-2

-

Q u (x) ²) w (s) ds —

(Q u (s) ² - Q u (x) ² ) w (s) ds +

+2

[x Qu (x)3 e

-2

-2

-

-

Qu (x) ²) w (s) ds

Qu (x)-2) w (s) ds = ^ Ii (x), i=1

where

Ku (x) = (Qu (x) + H) [Лu (x)]-1, and

GQu(x) (do, ui, f) (x) = —Au (x) Lqu(x) (x, f) — Au (x) <^{xQ (x)} [(Лu (x))^-1do+ +K_u(x) e^Q^и^ — A_u (x).¹¹ 'x ^Q ■^(x)[(I — K_u (x) e^2Q'^(x)) u1 — (Л_u (x))^-1e^Q(x)do\ .

Proposition 1. Assume (5) - (10). Then there exists w* > 0 such that for all w > w* :

IIP^ IIl(Lp(0,1;E)) — 2•

Proof. See [1, Proposition 2, p. 25].

□

Therefore for all w > w*, IP_UIl(_LP(_{0 1;}E)) — ^ which leads us to invert I + P_u in the space L (0,1; E) •

We can write for all w > w * and x E [0,1]

u (x) = Au (x)^-1(I + Pu)-¹G(x) (do, u 1, f) (x) •                    (12)

4.    Regularity of the Solution

Throughout this section we assume that w > w₁^*•

• 4.1.    Regularity of the Second Member GQ^(x) (d0, uE f)

For convenience we present the results below in the form of lemmas.

Lemma 7. Assume (5) — (7) and f E Lp (0,1; E) with 1 < p < to. Then for all w > w*

• 1) t -^ Qu (x) Jo e( ^sQ ^(x)f (s) ds E Lp (0,1; E);

• 2) t -^ Qu (x) Jt¹e^(s-t)Q“^(x)f (s) ds E L (0,1; E);

• 3) t -^ Qu (x) J1 e^(t+s)Q(x)f (s) ds E L (0,1; E);

• 4)    Jo esQ ^(x)f (s) ds E (D (Q (x)) ,E) 1 and J0 e^(1-s)Q ^(x)f (s) ds E (D (Q (x)) ,E) 1 . _p,p                                                                  _p,p

Proof. See [30].

□

We have the following lemmas as in [31]

Lemma 8. Fix x E [0,1], p E ]1, to[ and w > w*. Then

• 1)    t I—>A_u (x) e^tQ“^(x)^ E L^p (0,1; E) if and only if ^ E (D (A (x)) ,E) 1 ; 2p’^p

• 2)    t 1—>Q_u (x) e^tQ“^(x)^ E Lp (0,1; E) if and only if ^ E (D (A (x)) , E) 1 1 . 2p ⁺ 2 ^p

In the following, it is important to note that

(D (A (x)), E) x c D (Q (x)) C (D (A (x)), E) j. 2p^,p                                      2p +2 ^p

This is due to the reiteration property

• i)    (D (A (x)) ,E) 1 =(E.D (A (x))), J. = (E.D(Q (x)²)^ ± = 2p^{,p                          1}2p^{,p                            1} 2p^,p

= (E, D (Q (x))) x = L £ E : Qp £ (E, D (Q (x))) i V

• ² 2p^,p      I.                                               ¹ 2p’^p J

• ii)    (D (A (x)) ,E) J. ,  =(E,D (A (x))) 1  , = (E,D (Q (x)²)) 1  1

2p + 2 ,^p                        2  2p,^p                           2  2p ,^p

= (E.D (Q (x)))₁ - 1 = (D (Q (x)) ,E) 1 . _p,p                           _p,p

Proposition 2. Assume (5) - (11) and f £ Lp (0,1; E) with 1 < p < to. Then for all w > w*, x I—>Gq_u(x) (do, U1, f) (x) £ L^p(0,1; E) if and only if

(Q^ (0) - H)^-1do £ (D (A (0)) ,E) i

2p^,p

U1 £ (D (A (1)) ,E)x

2p^,p

Proof. Let x £ [0,1] and w > w*. We have

Gq-(x) (do, U1, f) (x) = Au (x) e^xQ(x)(Лu (x))^-1do + Au (x) e'^{1 x Q} '^(x)U1

-

-2Qu (x) e^xQ“^(x)Ku (x) [

e^sQ“^(x)f (s) ds + |q_u (x) e^{(1 x)Q-(x)}

-¹Qu (x) / e^{(x s)Q}“^(x)f (s) ds

²Jo

-1 Qu (x) f

^ e^(1-s)Q-^(x)f (s) ds

-

^Q(x)f (s) ds₊r (x,do,u1,f ) =

= Au (x) exQ“(x) (Лш (x))-1 do + Au (x) e(1-x)Q“(x)U1 + E Ji (x) + R (x, do, U1, f), i=1

where

R (x, do, U1, f) = Au (x) e^xQ“^(x)Ku (x) e^Q(x)U1

-

Au (x) e'^{1 x Q} '^(x)Ku (x) e^2Q-(x)U1

-

-Au (x) e     ^Q(x)(Лu (x))

¹■ " ^(x)do + 2 Qu (x) e^{xQ (x)}

+2Qu (x) e    ^Q(x)Ku (x) £

Ku (x) / e^(2-s)Q“^(x)f (s) ds+

-

¹Qu (x) e     ^Q(x)Ku (x)

2_o

e^(1+s)Q-(x)f (s) ds

-

e^(3-s)Q“^(x)f (s) ds.

For any £ £ E, k £ N, we have e^Q“^(x)£ £ D ^Q (x)^kj), so

Au (x) eQ(x)eQ(x)£ = e'Q“(x) Au (x) eQ-(x)£, and s 1—> Au (x) esQ-(x)eQ“ (x)£ is bounded and thus in Lp (0,1; E). To conclude it is enough to remark that Au (.) R (., do, u1, f) can be written as a sum of terms PAu (x) eQ(x)eQ“(x)Au (x), PAu (x) e1 ■ ' Q ■ (x)eQ“(x)Au (x), where P £ L (E), £ £ E.

For J₃, we consider the following problem:

( p' (x) - Qu (x) p (x) = f (x),   .x £ (0,1),

I p (0) = 0.

Let ф be the strict solution of problem (13). Fix x E [0,1], and set v (s) = e(x-s)Q“(x)ф (s), s E [0,x].

Then for each s E [0,x], we have v‘ (s) = —Qu (x) ex s Q (x)ф (s) + e,x s Q ■(x) [Qu (s) ф (s) + f (s)] = = Qu (x) e Q(x) [Qu (x)-1 — Qu (s)-1] Qu (s') ф (s) + e(x-s)Q“(x) f (s').

Integrating over ]0,x[ and applying Q_u(x) to both sides, we get:

= QQ( (x)²e^(x-s)Q“^(x)[Qu (x)^-1 0_x

= [ Qu (x)² e^(x-s)Q“^(x)[Qu (x) 0

—

Qu (x) ф (x) =

Qu (s)¹] Qu (s) ф (s) ds + Qu (x) /

J 0

e⁽s)(Q^(x)f (s) ds =

-1

— Qu (s)¹] Qu (s) ф (s) ds + J3 (x);

see [26, p. 56, 57]. Due to [32, Theorem 5.11, p. 59], we have x 1—> Qu (x) ф (x) in Lp (0,1; E) and due to Lemma 7 and Lemma 8, we have x 1—>

/ x Qu (x)² 0

e(x-s)Q„ (x)

[Qu (x)^-1— Qu (s)^-1] Qu (s) ф (s) ds,

in Lp (0,1; E). Then x 1—>J₃(x) is in L (0,1; E).

The same technique is used for the other terms. Therefore, due to Lemma 5, we can write:

Au (x) exQ(x) (Au (x))-1 do + Au (x) e(1-x)Q“(x)U1 = Au (x) extQ(x) (Qu (x) — H)-1 do+ +Au (x) exQ(x) (Qu (x) — H)-1 W (x) d0 + Au (x) e(1-x)Q^(x)u1, where W (x) E L (E) and R (W (x)) С П^ D (

2ni

Au (x)   Q ^(x)(Qu (x) — H)^-1do + Au (x) e^(1-x)Q“^(x)U1 =

= Au (x) exQ^(x)(Qu (x) — H)^-1do +Au (x) ex^^(x)(Qu (0) — H)^-1do +A_u(x) e^(1-x)Q^^(x)u₁

—

—

—

Au (x) Q^(x)(Qu (0) — H)^-1do+ Au (0) ■⁽⁰⁾(Qu (0) — H)^-1do+ Au (1) e^(1-x)Q“⁽¹⁾U1+

2ni re 1

+Au (0)   Q (o) (Qu (0) — H)-1 do + Au (1) e(1-x)Q“(1)U1 = e ^Au (x) (Au (x) — z)-1 [(Qu (x) — H)-1 — (Qu (0) — H)-1] dodz—

'    [Au (x) (Au (x) — z)^-1— Au (0) (Au (0) — z)^-1] x (Qu (0) — H)^-1dodz—

2ni

I e-^-x [Au (x) (Au (x) — z)^-1— Au (1) (Au (1) — z)^-1] U1dz+ ^Г+Au (0) e^(o)(Qu (0) — H)^-1do + Au (1) e^(1-x)Q“⁽¹⁾u_b

Using the algebraic identity:

Au (x) (Au (x) — z)^-1— Au (0) (Au (0) — z)^-1=

=  zAu (x) (Au (x) — z)-1 [Au (x)-1 — Au (0)-1] Au (0) (Au (0) — z)-1, we obtain:

A. (x) e^xQ(x)(Qu (x) - H)^-1do + A. (x) e ■ ^{x Q} '^(x)ui =

1 [ e '•     A (x) (A. (x) - z)^-1[(Qu (x) - H)^-1- (Qu (0) - H)^-1] dodz-

2ni L

-Л I z '   A. (x) (A. (x) - z)^-1[A. (x)^-1- A. (0)^-1] X

2ni Jr xA. (0) (A. (0) - z)-1 (Qu (0) - H)-1 dodz -     [ z 2 A (x) (A. (x) - z)-1 x

2ni Jr x [A. (x)-1 - A. (1)-1] A. (1) (A. (1) - z)-1 U1dz + A. (0) exQ“(o) (Q. (0) - H)-1 do+

+A_u(1) e^{(1 x)Q}“⁽¹⁾u1 = a1 (x) + a₂ (x) + a₃(x) + a₄(x) + a₅(x).

For a1 (x), we have

lai (x)|e< C f e^-Colzl1^/2xx^a+2^ |dz| We<

<C

e-σxα+2µ

2^ ||<Ме<      •    ||do|e .

x

Then

(J0 llai ^(x)|pEdx

x^(a+2M LPdx^ Id0I_E< +w,

so

x I—>a1 (x) G Lp (0,1; E).

For the second term, we have:

α

|a2 (x)|_E< C I Ie e^-c0^|z|/ ^x| A. (0) (A. (0) - z)^-1(Q. (0) - H)^-1dodz| d |z| < г                        Izl

< с У |z| e < C У |z| e

----------1

----------1

c^0|z|1/2xX |Q. (0) (A. (0) - z)^-1Q (0) - H)^-1Q^ (0) dodz 11 d |z| < |z|

α c0|z| / x| (A. (0) - z)-1 Qu (0) (Qu (0) - H)-1 Qu (0) dodz| d |z| < |z|

α

< C   e-co|z| xП? |dz||Qu (0) do»E< г                   lzl

<C

e

_ax^a2ada

^σx2

x²

||Q. (0) do^E < Cx^a+2p-2 IQ (0) do^E .

Then

1 x(α+2µ

2^)pdx^ |Q^ (0) do|_E< +w,

we conclude that:

x I—>a₂(x) G Lp (0,1; E).

The same technique is used for the other terms. Finally

^{x 1}—>GQ^(x) (do,^u1, f) ^{(x) G} Lp^{(0,1; E}),

if and only if

/ x -^ A (0) e Q ^(o)(Qu (0) - H)^-1do G Lp (0,1; E),

1 x I—> Аш (1) e<1-x)Q“(1)ui G Lp (0,1; E), which is equivalent, by Lemma 8, to

( (Qu (0) - H)^-1do G (D (A (0)) ,E)^p,

] ui G (D (A (1)) ,E)x

I                                2p,p